I I have a shot where the E20 focuses very sharp at the left edge, but not at the foreground, whereas the S19 focuses sharp on the foreground, but not at the (very) left edge.

Sound like wrongly focal point between each lens swap

DOF does not explain this. Camera (tripod?) and possibly decentering perhaps, but for now, I am betting on in-camera lens corrections that may have a hand in this (straightening of lines), and are caused by the different optical formulas of both lenses.

You should have DISABLE IN CAMERA CORRECTION to reduced outside factor (cpu/in camera software corner sharpening) that is ruining your test results

You are just assuming that that is the problem. Unless the distortion is very extreme, I don't think it would be so severe. It could be extreme field curvature, not all unlike the 16mm and some other lenses.

Let the chips fall where they may. Don't HIDE or MAKE EXCUSE for Sony. Please show each lens "raw & uncorrected" image. This is the only way we can compare a lens optics.

It would help to determine whether or not it is contributing to any artifacts, but it has nothing with hiding or making excuses. In its final form, people will want to use the lens after corrections, so at the end of the day, that is what needs to be compared.

Otherwise, we never know if Sony is sharper due to in-camera-corner-sharping as part of its software corrections, or as your own result show, make it worst and produce mushy corners.

But at the end of the day, it doesn't matter a lot why it is, as that is the way it is. It's not like we're going to hack the lens and make it work differently.

Either way, your test is ruin by enabling the in-camera-correction

Hardly, the test is correct as far as in-camera correction-- it's testing in the way we would use it.

Now, there may be other things about the test that are wrong. (Where was it focused? Were all of the settings the same or was something forgotten?)

Aside from this strange quirk, I find the E20 and S19 fairly comparable. The E20 does seem a tad softer at the corners, but it seems sharper in the center.

I cannot say which is lens is sharper because as you indicated, CDAF shoot different AF points after each lens swap. The only valid way to compare lens is to MANUAL FOCUS with peaking. This way, you take the "AF ERROR" out of the testing equation

But if one tends to use AF, it would be good to note AF errors if they are common. So here again, there is value in testing as-shot.

Now, if Henry is willing to do FURTHER testing, then, yes, manual focusing might help determine if it really is back-focusing too much. But without his test, how would you know that there was an issue in the first place?

(Is this difference something to do with PDAF vs. CDAF?)

I'll look forward to your re-shoot. The current results are invalidated and I cannot form any conclusion when I doubt the test results itself.